Figure 9 shows the situation where we do 1:1 imaging from the source into the fiber. From the Gaussian lens equation, we have s_{1} = s_{2} = 2f in this configuration. Since y_{2} is simply the radius of the receiving fiber, the light is collected only from an area of the source of radius y_{1} = y_{2}. The maximum angle coupled into the fiber is restricted by θ_{2} = NA. Since this is a 1:1 imaging problem, the maximum angle sampled from the light source is then also θ_{1} = NA. Thus, it is the fiber and not the lens that puts the restriction on how much light is collected. As long as we choose a lens with sufficiently small f/#, that is f/# <1/(2NA), then we will collect the maximum possible amount of light from the source into the fiber.

Note what happens if we remove the lens and move the fiber directly against the source. That is, we **butt couple** the fiber to the source. In that case, the fiber collects light from an area of radius y_{2} and angle NA. But, this is exactly what we accomplished in our 1:1 imaging. An imaging system collecting light from a diffuse source into a fiber cannot collect more than could be collected by butt coupling.

In fact, this is a completely general result. Consider a configuration more like Figure 7, where we might choose, for example, to condense light onto the fiber end face from an area of the source with y_{1} = 5y_{2}. We are now collecting light from an area that is 25x larger than the area of the fiber. However, the optical invariant tells us that we must now have θ_{1} = NA/5, so we are collecting light from a solid angle that is reduced by the same factor of 25. Thus, the total light that is collected, irradiance x area x solid angle, is constant. What a lens system can achieve is only to retrieve the efficiency of butt coupling when the fiber must be placed at a distance from a diffuse source. Therefore, for maximum efficiency, choose a fiber with the largest possible core diameter and the largest available numerical aperture.

When we need to couple laser light into a single-mode fiber, we move from the ray optics picture in which we have worked to this point to a Gaussian mode-matching problem. This application is included here for completeness in discussing coupling light into optical fibers.

In order to couple light of wavelength λ from a collimated laser beam of 1/e2 diameter D into a fiber of mode field diameter ω, choose a lens with a focal length

f = D( πω/4λ)

Let’s consider the situation where we use a Newport F-915 fiber coupler to couple light from a R-30992 laser (λ = 633 nm, D = 1.2 mm) into a Newport F-SV optical fiber (ω = 4.3 µm). We find f = 6.4 mm. For this application, use the Newport M-20X objective, f = 9 mm, as the closest fit to the correct focal length.

The coupling efficiency depends upon the overlap integral of the Gaussian mode of the input laser beam and the nearly Gaussian fundamental mode of the fiber. This overlap integral is the same whether the input mode is the larger or the smaller of the two modes. The focal length of the M-20X is too large by a factor of 1.4 while the focal length of the M-40X too short by a factor of 0.7, so the M-20X will be the better fit for this application.